1. 5.6 Solving Quadratic Equations
Quadratic Equations are equations of 2nd degree. In other words, they have an x2 term in them.
If AxB=0 then A=0 or B=0
This is the basis for the approach we use to solve quadratics in 5.6.

2. X2 + 5x + 6 = 0 Set = 0
(x + 3)(x + 2) = 0 Factor
X+ 3 = 0 OR x + 2 = 0 set each factor = 0
solve for x
X = -3 OR x = -2

3. X2 = 4x Set = 0
X2 - 4x – 21 = 0 leading coeff +
(x - 7)(x + 3) = 0 Factor
X - 7 = 0 OR x + 3 = 0
set each factor = 0
X = 7 OR x = -3 solve for x

4. 4X2 = 25
4X = 0
(2x - 5)(2x + 5) = 0
2x - 5 = 0 OR 2x + 5 = 0
2x = 5 OR 2x = -5
X=2 ½ OR x = -2 ½

5. (x - 2)(x - 1) = 12
x2 - 3x + 2= 12
x2 - 3x – 10 = 0
(x - 5)(x + 2) = 0
x - 5 = 0 OR x + 2 = 0
x = 5 OR x = -2

6. 6.1 Simplifying Rational Expressions
Remember what a rational number is? Any number that can be written as a fraction.
Rational Expressions then are expressions that are in fraction form.
Remember that it is illegal to divide by zero so much like we cannot have zero on the bottom of a numeric fraction, we also cannot have zero on the bottom of an algebraic fraction.

7. In this rational expression, x=2
To find out what values are excluded for x, set the denominator = 0 and solve for x. This may require you to factor first. When you solve, this tells you what values would make the fraction undefined. These values are exclusions for x.
X - 2 = 0
X = 2

8. there are no exclusions on the x
x - 1 = 0 so x=1
x2 – 4 = 0
(x+2)(x-2)=0 so x=2 and x=-2

9. Just a reminder: a negative fraction can have the negative on top, bottom or out front. BUT not top and bottom.
Simplify a rational expression using the quotient rule:
14x2y5z 1y4
28x5yz2 2x3z

10. Simplifying rational expressions by factoring
Factor top and bottom
Cancel out any common factors
ab – b2 b(a-b) a-b
2b b 2
5x3 + 10x2 – 25x 5x(x2 + 2x - 5) (x2+2x-5)
10x x x

11. X2 + 2x – 3 (x + 3)(x - 1) x - 1
x x + 3
r2 – 25 (r + 5)(r – 5) r +5
r – 5 r – 5
3x2 – 10x – 8 (3x + 2)(x – 4) 3x +2
X2 +3x – 28 (x + 7)(x – 4) x + 7

12. Watch out for this!
(3x – 7) -(-3x + 7)
(7 – 3x)(5 – x) (7 – 3x)(5 – x)
5-x -(5 – x) 5 - x

13. 6.2 Multiplying and Dividing Rational Expressions
Remember how to multiply and divide fractions?
Mult: top times top; bottom times bottom and reduce
Divide: leave the first fraction; flip the second and multiply and then reduce

14. To multiply or divide rational expressions, you will do the following steps:
Factor top and bottom of each fraction
Flip if you are dividing
Cancel out any common factors
Multiply what’s left over

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